When a story has two unknowns, you usually need two equations, one for each fact the story gives. This puts together writing equations from words and solving systems.
Why it works
One equation with two unknowns, like $a + s = 10$, has endless solutions: $(1, 9)$, $(2, 8)$, and so on. A second, independent fact, like how much money was collected, rules out all but one. That is why each equation must say something different. Two equations that say the same thing (like $a + s = 10$ and $2a + 2s = 20$) do not pin anything down.
- Name both unknowns, with units.
- Write one equation per fact. Many problems have a count (or total amount) equation and a value equation.
- Solve by elimination or substitution.
- Answer the question in words and check both facts in the story.
Key idea
Two unknowns need two independent equations. Say what each variable means, write one equation per fact, solve, and check against the story.
Worked examples
Worked example: Tickets
A club sold $10$ tickets for \$65. Adult tickets cost \$8 and student tickets \$5. How many of each were sold?
- Let $a$ = adult tickets and $s$ = student tickets.
- Count: $a + s = 10$. Money: $8a + 5s = 65$.
- Multiply the count equation by $5$: $5a + 5s = 50$. Subtract it from the money equation: $3a = 15$, so $a = 5$.
- Then $s = 5$. Check: $8(5) + 5(5) = 65$.
Answer: $5$ adult and $5$ student tickets.
Worked example: Current
A boat goes $36$ miles downstream in $2$ hours and returns upstream in $3$ hours. Find the boat's speed $b$ and the current's speed $c$.
- Downstream speed is $b + c$; upstream is $b - c$.
- $2(b + c) = 36$ gives $b + c = 18$. $3(b - c) = 36$ gives $b - c = 12$.
- Add: $2b = 30$, so $b = 15$ and $c = 3$.
Answer: Boat $15$ mph, current $3$ mph.
Common mistake
Writing two equations that say the same thing, or never saying what each letter stands for. If you cannot say in words what fact each equation records, check the story again: each sentence with a number usually gives one equation. Defining $a$ and $s$ with units also prevents mixing up which answer is which.